Chapter 13 Answers A.69 Chapter 13 Answers 13.1 Vectors in the Plane 31. y 1. (4,9) 3. (-15, 3) 3 5. Y = 1 7. No solution 9. No solution 11. No solution 13. a = 4, b = - I 15. a = 0, b = I 17. (XI, YI) + (0,0) = (XI + 0, YI + 0) = (XI' YI) 3 x 19. [(XI' YI) + (X2, Yz)] + (x3' YJ) = (XI + X2 + X3, YI + Yz + Y3) = (XI' YI) + [(X2, Yz) + (x3, Y3)] 21. a(b(x, Y» = a(bx, by) = (abx, aby) = ab(x, y) 33. y 23. y 2 • • • • •• • • • • •• x x • •• • • • 4 • • • • • • 35. 25. (a) k(l,3) + 1(2,0) = m(1, 2) (b) k + 21 = m and 3k + 0 = 2m x (c) k = 4,1 = 1 and m = 6; i.e. 4S03 + S2 = 6S02 27. (a) d (b) e 29. (a) c + d = (6,2) -6 y 37. (a) 2 .r Q x 3 6 vAw Pl....-).R u 234 x (b) -2e+a=(-1,0) (b) v=(l,2); w=(1, -2); u=(-2,0) (c) 0 y 39. (a) y -2e+a x x (b) (0, I) A.70 Chapter 13 Answers (c) (0,5/2) 9. (d) (0, -2) x 11. w -2 (e) (I, y) (f) v=(O,y) 41. (a) Yes (c) Eliminate rand s. w (b) v = -(s/r)w (d) Solve linear equations. 13. - i + 2j + 3k 15. 7i + 2j + 3k 13.2 Vectors in Space 17. i - k 19. i - j + k 21. i + 4j, B ~ 0.24 radians east of north 1. 23. (a) 12:03 P.M. (b) 4.95 kilometers 25. .\" F = 50 lb. F, = 50 sin (50') lb ~ 38.3 lb. 3. (3. 27. The points have the forms (O,y,O), (O,O,z), (x, y, 0), and (x, 0, z). 29. y y 5. (11,0, II) 7. (-3, -9, -15) Chapter 13 Answers A.71 31. 9i+ 12j+ 15k 33. 26i + 16j + 38k 33. (I/13)i + (I/13)j + (1/13 )k, (1/ Ii)i + (1/ Ii)k 35. a = !, b = - ! 37. a = 5, b = 2 35.13 39. (4.9,4.9,4.9) and (-4.9, -4.9,4.9) newtons. 37. Ii 41. (a) Letting x, y, and z coordinates be the number 39. (i) ~14t2 - 121 + 4 of atoms of C, H, and 0 respectively, we get (ii) t = 3/7 p(3,4,3) + q(0,0,2) = r(l,0,2) + s(0,2, I). (iii) ~1O/7 (b) P = 2, r = 6, s = 4, q = 5 41. 13 knots (c) 0 43. Solve one equation for I and substitute. The line is vertical when XI = X2' 2(3.4.3)+ 5(0.0.2) = 6(1.0,2) + 4(0,2, I). 45. When the angle between the vectors is O. 6(1,0.2) 4(0,2. I) 13.4 The Dot Product 1.4 3. 0 10 H 5. ~ 0.34 radian 7. 7T /2 radians 9. (l/{S)i + (2/{S)j 10 11. 0.955 radians c 43. (a) P~I=(-I,-I,-I),Po=(I,O,O), PI = (3, I, I), P2 = (5,2,2) (b) x (i+j+k) .\' 13. Use Figure 13.4.2. 15. 142 17. '150/11 19. x + Y + z = 0 21. x = 0 (c) The line through (I, 0, 0) parallel to the vector 23. - x + y + z - I = 0 (2, I, I). 25. (2/ff4)i + (3/ff4)j + (l/ff4)k 27. (I/Ii)i + (1/Ii)j 29. x + Y + z - I = 0 13.3 Lines and Distances 31. x = I + t, Y = I + I, z = I + I 1. Use vectors with tails at the vertex containing the 33. (I, - 1/2,3/2), ff4 /2 two sides. 35. x = (22 - 9t)/7, Y = (- 6 - 2t)/7, z = I 3. Use the distributive law for scalar multiplication. 37. 313. 5. (I, - t) 39. Ii /2. 7. x = I - I, Y = I - I, z = I 41. Letting (a,b) and (c,d) be the given points, the 9. x = t, Y = t, z = t equation of the line is (a - c)x + (b - d)y 11. x = I - t,y = I - I, z = I = (l/2)(a 2 - c2 + b2 - d 2). Use this to show that 13. x = - I + 3t, y = - 2 - 2t the two points are equidistant from points on the 15. (-2, -J,O) 17. No line. 43. (a) 3 19. 13 21. Ii (b) -2 23. 21i 25. ::!:: Ii46 (c) 213 27. 112i + j + 2kll = 3, which is less than 13 + Ii· (d) 3 29. One solution is u = i, v = - i, w = i. 45. Letting PI = (p,q) and P2 = (r,s), we have 31. Each side has length Ii. (r - p)x + (s - q)y = (r2 + s2 _ p2 - q2)/2. A.72 Chapter 13 Answers 47. To show that v and w = (v . el)el + (v . e2)e2 are 11. Compute the two determinants. equal, show that v - w is orthogonal to both el 13. Compute the two determinants. and e2' 15. 0 19. -6 49. FI = - (F12)(i + j) and F2 = (F12)(i - j) 17. 4 21. 9 51. (a) F = (3v'2 i + 3v'2 j) 23. abc (b) ~ 0.322 radians j (c) 18v'2 25. - I ~1=-i-3j+3k 53. Use the component formula for the dot product. Ii I 55. (a) [12.5)2 + (16.7)2 - (20.9)21/[(12.5)(16.7)] IS j close to O. 27. I ~I=i-j+k (b) 0.54% Il I 57. (a) Let s = tVa2 + b2 + c2 j 29·1 0 1= -2j (b) Use the fact that Ilull = I. ~ 0 -> 2 (c) Use IIull = I. 31. 6 (d) For LI and L 2 , 33. 12 coso: = 1//3 so 0: = cos-I(l//3), 35. Compute and simplify. cos f3 = 1//3 so f3 = cos-I(I//3), 37. Compute both determinants and compare. cos y = 11/3 so y = cos-I(ll /3). 39. Use Example 8 after renumbering the vectors. For L3 and L 4 , 41. coso: = 11m so 0: = cos-I(l/m), cos f3 = 11m so f3 = cos-I(l/m) and cosy = 91m so y = cos-I(9Im). (e) Only the line t(l, I, I). 13.5 The Cross Product I.j+k 3. 2i - 2j + 4k 5.9i+18j 7. 6i - 2k 9. -i + k 11. 3v'2 13. 2 15. -(l/v'2)j + (1/v'2)k 17. - (I I v'2)i - (I I v'2)j 19. (v'2/6)i - (v'2/6)j + (2v'2/3)k 21. 2x + 3y + 4z = 0 23. x - 3y + 2z = 0 25. 3v'2/2 27. The points are collinear, so the area is zero. 29. Substitute component expressions for VI and V2' 31. The angle between the vectors is () - 1/1. Now use property I in the box on p. 679. 33. Use the result of Exercise 32. 35. Show that M satisfies the defining properties of RxF. 37. Show that nl(N X a) and n2(N X b) have the same magnitude and direction. 39. (a) Draw a figure showing the two lines and the plane in the hint. (b) v'2 41. If F is the gravitational force, the gyroscope ro­ tates to the left (viewed from above). 43. Substitute the expressions for x and y in the equa­ 13.6 Matrices and Determinants tions. 45. Substitute the given expressions for x, y, and z in 1. 2 3.0 the equations. 5. -2 7. 25 47. x=37/13,y= -3/13 9. ac 49. Compute both determinants and compare. Chapter 13 Answers A.73 51. Subtract four times row 1 from row 2, subtract 49. This is a (double) cone with vertex at the origin. seven times row 1 from row 3, expand by column I and then evaluate the 2 X 2 determinant. 53. 3, -6 Review Exercises for Chapter 13 y 1. (2,8) 3. (- 1,,- 2, 17) 5. Iii + j - k 7. -4i + 7j - 11k 9.6 11. -2k 13. i - 2j 51. (a) Draw a vector diagram. (b) Use c X c = O. 15. 2i + j - 3k (c) Use part (b). 17. (-2//IT)i + (3//IT)j + (3//IT)k 53. Use the dot product to show that the vectors a - b 19. 0 (the three vectors lie in a plane). and - a - b are perpendicular. 21. (a) (6,6) 55. 3 57. 1 59. -2 61. 0 63. y38T 65. 29/2 4 (b) 1/3 69. Use the fact that IIall 2 = a' a, expand both sides and use the definition of c. 71. x=3/7,y= -29/21,z=23/21 73. -162 75. Each side equals 2 2xy - tyz + 5z 2 - 48x + 54y - 5z - 96. (Or switch the first two columns and then subtract the first row from the second.) 77. v is orthogonal to i (b) (9,7) 79. (a) 4k 23. Put the triangle in the xy-plane; use cross products (b) 20,12 i + 20,12 j with k. 81. (a) Substitute i, j, and k for w. 25. (1825 - 600y''2) I /2 R:: 31.25 km /hr. (b) (u - v) . w = O. 27. (a) 70 cos B + 20 sin B (c) Repeat the reasoning in (a) (b) (21 If + 6) ft.-lbs. (d) Apply (c) to u - v. 29. x = 1 + t, y = 1 + t, Z = 2 + t 83. (a) 31. x = 1 + t, y = 1 - t, Z = 1 - t 33.